I have finally been able to confirm that there was a brand of cigars named El Verso (not Alverzo); however, it doesn't have the word "Don" connected to it. (There are/were several other brands that do/did begin with "Don": Don Corello, Don Equestro, Don Gudo, Don Ovando, Don Rodrigo, etc. "Don" is a Spanish honorific.)

It now seems unlikely this has any connection to "Don Alverzo's tweezers."